# DiggleGratton: Diggle-Gratton model

### Description

Creates an instance of the Diggle-Gratton pairwise interaction point process model, which can then be fitted to point pattern data.

### Usage

 `1` ``` DiggleGratton(delta=NA, rho) ```

### Arguments

 `delta` lower threshold δ `rho` upper threshold ρ

### Details

Diggle and Gratton (1984, pages 208-210) introduced the pairwise interaction point process with pair potential h(t) of the form

h(t) = ((t - δ)/(ρ - δ))^κ, { } δ ≤ t ≤ ρ

with h(t) = 0 for t < δ and h(t) = 1 for t > ρ. Here δ, ρ and κ are parameters.

Note that we use the symbol κ where Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987) use β, since in spatstat we reserve the symbol β for an intensity parameter.

The parameters must all be nonnegative, and must satisfy δ ≤ ρ.

The potential is inhibitory, i.e.\ this model is only appropriate for regular point patterns. The strength of inhibition increases with κ. For κ=0 the model is a hard core process with hard core radius δ. For κ=Inf the model is a hard core process with hard core radius ρ.

The irregular parameters δ, ρ must be given in the call to `DiggleGratton`, while the regular parameter κ will be estimated.

If the lower threshold `delta` is missing or `NA`, it will be estimated from the data when `ppm` is called. The estimated value of `delta` is the minimum nearest neighbour distance multiplied by n/(n+1), where n is the number of data points.

### Value

An object of class `"interact"` describing the interpoint interaction structure of a point process.

\spatstatAuthors

### References

Diggle, P.J., Gates, D.J. and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction point processes. Biometrika 74, 763 – 770.

Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.

`ppm`, `ppm.object`, `Pairwise`

### Examples

 `1` ``` ppm(cells ~1, DiggleGratton(0.05, 0.1)) ```

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